Forecasting

 

 

1. Basic Forecasting Tools

 

 

1.1 Forecasting Methods and Examples

 

1.1.1 Examples:

 

The first example, [Web: Australian Monthly Electricity Production ], displays a clear trend and seasonality. Note that both the seasonal variability as well as the mean show a trend.

 

The data [Web: US Treasury Bill Contracts ] shows a trend, but there is less certainty as to whether this trend will continue.

 

The data on [Web: Australian Clay Brick Production ] contains occasional large fluctuations which are difficult to explain, and hence predict, without knowing the underlying causes.

 

Exercise 1.1: Make Timeplots of each data set: Australian Monthly Electricity, US Treasury Bills, Australian Clay Brick.

 

1.1.2 Quantitative and Qualitative Approach:

Quantitative approach relies on sufficient reliable quantitative information being available. Qualitative approach is an alternative if expert knowledge is available.

 

1.1.3 Explanatory Versus Black-Box Models:

An explanatory model is one that attempts to explain the relationship between the variable to be forecast and a number of independent variables. eg

 

 

GNP = f(monetary and tax policies, inflation,

 capital spending, imports, exports) + Error

 

A time series model is one that attempts to relate the value of a variable(s) at one time point with values of the variable(s) at previous time points. eg

 

          GNPt+1 = f(GNPt, GNPt-1, GNPt-2, ....) + Error

 

A black-box model is one that simply tries to relate future values of the variable of interest to previous values, without attempting to explain its behaviour in terms of other variables.

 

Thus simple time series models, like the one above, are 'black-box'.

 

More complex time series models are explanatory in that they try to relate the value of the variable of interest not simply with its previous values but also with previous values of other 'explanatory' variables.

 

         

1.2 Graphical Summaries

 

1.2.1 Time plot.

Always make a time plot and look for patterns:

    (i) A time series is said to be stationary if distribution of the fluctuations is not time dependent  In particular both the variability about the mean, as well as the mean must be independent of time.

    (ii) A seasonal/periodic pattern is one with a yearly, monthly or weekly period.

    (iii) A cyclical pattern is one where there are rises and falls but not of regular period.

    (iv) A trend is a long term increase or decrease in the variable of interest.

eg  [Web: Australian beer production: Time Plot ]

 

1.2.2 Seasonal plot.

A seasonal plot is one where the time series is cut into regular periods and the time plots of each period are overlaid on top of one another.

eg  [Web: Australian beer production: Seasonal Plot ]

 

Exercise 1.2: Produce time and seasonal plots for the Australian beer production  data.

 

1.2.3 Scatterplots.

This plots the relationship between two variables, but does not necessarily have to have time as one of the variables.

eg [Web: Price/Mileage relationship for 45 cars ]

 

Exercise 1.3: Produce a scatterplot for the Price/Mileage relationship for 45 cars data.

 

 

1.3 Numerical Summaries

 

1.3.1 Statistics.

A statistic is a summary quantity calculated from a data set.

 

1.3.2 Univariate Statistics.

Commonly used statistics are the mean, median, deviation, mean absolute deviation (MAD), variance or mean square deviation (MSD); standard  deviation (SD).

 

These are calculated for the data:[Web: 19 Japanese Cars ]

 

EXCEL contains several of these statistics as Worksheet Functions, specifically:

AVERAGE, MEDIAN, VAR, STDEV.

Note: VAR and STDEV now use n – 1 in the divisor. [Also they use an old fashioned version of the formula, which is not fully robust.]

 

Exercise 1.4: Reproduce, in a spreadsheet the calculations made in the example:  19 Japanese Cars.

 

1.3.3 Bivariate.

The most commonly used statistics for bivariate data is the covariance, and the correlation coefficient. If we have n pairs of observations (Xi, Yi) on two variables X and Y then the formulas are:

 

and

         

 

The correlation coefficient is a standardised version of the covariance and its value is always between -1 and 1. Values close to each limit indicate a strong linear relation between the two variables.

eg These statistics are calculated for the data: [Web: 19 Japanese cars (bivariate) ]

 

EXCEL has the Worksheet Functions: COVAR, CORREL. However COVAR uses n in the divisor. Why does it not matter whether n or (n-1) is used for the correlation?

 

Exercise 1.5: Calculate these statistics for yourself, and using the Worksheet functions: 19 Japanese Cars.

1.3.4 Autocovariance; Autocorrelation.

The use of covariance and correlation can be extended to a time series {Yt}. We can compare Yt with the previous lagged value Yt-1. The autocovariance, ck, and autocorrelation at lag k, rk, are defined as

 

                  

and

         

 

The complete set of autocovariances is called the autocovariance function, and the set of autocorrelations, the autocorrelation function (ACF).

 

Exercise 1.6: Calculate the ACF for the Australian Beer Production data (ACF). [Web: Australian Beer Production ACF. ]

 

Note that there is a peak at lag 12 and a trough at lag 6. It is not usual to plot more than n/4 lags, as the number of terms in the summation being relatively small, means that the estimates of the correlations for large lags are correspondingly less reliable.

 

Exercise 1.7: Write a VBA macro to calculate the autocorrelation function. The macro should have as input the column of n observations, and should output the autocorrelation function up to lag m = n/4.

 

 

 

 

1.4 Measures of Accuracy

 

1.4.1 Forecasting Errors

Let Ft be the forecast value and Yt be the actual observation at time t. Then the forecast error at time t is defined as

 

et = Yt - Ft.

 

Usually Ft is calculated from previous values of Yt right up to and including the immediate preceding value Yt-1. Thus Ft predicts just one step ahead. In this case Ft is called the one-step forecast and et is called the one-step forecast error. Usually we assess error not from one such et but from n values. Three measures of error are:

 

(i) The mean error

(ii) The mean absolute error

                            

(iii) The mean square error

                             .

 

The mean error is not very useful. It tends to be near zero as positive and negative errors tend to cancel. It is only of use in detecting systematic under or over forecasting.

 

The mean square error is a squared quantity so be careful and do not directly compare it with the MAE. Its square root is usually similar to the MAE.

 

The relative or percentage error is defined as

                            

The mean percentage error is

                            

and the mean absolute percentage error is

                            

 

Exercise 1.8: Set up NF1 and NF2 for the Australian Beer Data (NF1, NF2). [Web: Australian Beer Data (NF1,NF2).]

Calculate the ME, MAE, MSE, MPE, MAPE for the Australian beer series data using NF1 and NF2:

 

NF1:                    Ft+1 = Yt

 

This simply takes the present Y value to be the forecast for the next period.

 

The second naive forecast takes into account a seasonal adjustment. Suppose that the current time point is t = 12m + i where m is the number of complete years data available. Then, assuming no trend we can take the current monthly averages for j = 1,2,..., 12 as

                            

 

The second naive forecast is then

 

NF2:                    Ft+1 = Yt - Si + Si+1

 

Hint: The summation formula for Sj is not very convenient to enter directly on a spreadsheet. It is much easier to use an updating formula instead. It we write St for the seasonal index corresponding to time t (= 12m + i) (          i.e. t corresponds to ith month in the (m+1)th year), then

i.e.

.

 

This is the formula used in the spreadsheet.

 

 

1.4.2 ACF of Forecast Error.

It is often useful to regard the one-step forecast errors as a time series in its own right, and to calculate and plot the ACF of this series. This has been done for the Australian beer production series. [Web: Australian Beer Data (NF1,NF2).]

Notice that there is pattern in the series and this has been picked up by the ACF with a high value at lag 12. Do not read too much into the other autocorrelations as one should expect departures from zero even for the ACF of a random series.

 

1.4.3 Prediction Interval.

Assuming that the errors are normally distributed then one can assess the accuracy of a forecast by using  as an estimate of the error then an approximate prediction interval for the next observation is

 

 

where z is a quantile of the normal distribution. Typical values used are:

 

                                   z         Probability

                                1.282       0.80

                                1.645       0.90

                                1.960       0.95

                                2.576       0.99

 

 

1.5 Transformations

 

Sometimes a systematic adjustment of the data will lead to a simpler analysis. We consider just two forms.

 

1.5.1 Mathematical Transforms

There are two ideas that are helpful in selecting an appropriate transform.

 

First, it is usually easier to analyse a time series is the underlying mean is constant, or at least varies in a linear way with time.  Thus if the behaviour of the actual data has the form

 

                             Yt = at p + εt

 

where a and p are constants and εt is an random 'error', then the transform

 

                             Wt = (Yt)1/p = (at p + εt)1/p = bt + δt ,

 

where b = a1/p, makes Wt  look more 'linear' than Yt . Note that the transformed 'error', δt, will depend in a complicated way on εt, a, p and t. However in many situations the behaviour of δt will remain 'random' looking and be no more difficult to interpret that the initial error εt . The above is known as a power transform.

 

Another useful transform is the logarithmic transform:

 

                              Wt = log e (Yt).

 

This can only be used if Yt > 0, as the logarithm of a negative quantity is complex valued.

 

The second idea is that the random errors are most easily handled if their variability is not time dependent but remains essentially constant. A good transformation should therefore be variance stabilizing, producing errors that have a constant variance. For example if

 

                             Yt = a(t + εt) p

 

where the εt have a constant variance, then the power transform

 

                             Wt = (Yt)1/p =  a 1/p(t + εt) = bt + δt

 

where b = a1/p and δt = bεt will not only linearise the trend, but will also be variance stabilizing, as δt will have constant variance.

 

Finally note that, though we analyse the transformed data, we are really actually interested in the original sequence. So it is necessary to back transform results into the original units. Thus, for example in the last case, we might analyse the Wt and estimate b, by,  say  but we would back transform to estimate a by

 

.

 

An important but somewhat difficult technical issue is that such transforms can destroy desirable properties like unbiasedness. A well known case concerns a random sample X1, X2, ... Xn, of size n. Here, the sample variance given by the formula

 

                            

 

is known to be an unbiased estimator for the variance. However, s, the obvious estimator for the standard deviation is not unbiased. When n is large this bias is, however, small.

 

Exercise 1.9: Plot the  Australian Monthly Electricity data using the square root and the (natural) log transforms.

[Web: Australian Monthly Electricity Production ]

 

1.5.2 Calendar Adjustments.

If data is for calendar months, then account might have to be taken of the length of a month. The difference between the longest and shortest months is about (31- 28)/30 = 10%. The adjustment needed is

 

  

 

Exercise 1.10:: Make separate time series plots of Yt  and Wt for the data on the Monthly Milk production per cow.

[Web: Monthly Milk ]